TY - JOUR
T1 - The two-phase problem for harmonic measure in VMO
AU - Prats, Martí
AU - Tolsa, Xavier
PY - 2020/6/1
Y1 - 2020/6/1
N2 - Let Ω +⊂ Rn+1 be an NTA domain and let Ω -= Rn+1\ Ω +¯ be an NTA domain as well. Denote by ω+ and ω- their respective harmonic measures. Assume that Ω + is a δ-Reifenberg flat domain for some δ> 0 small enough. In this paper we show that logdω-dω+∈VMO(ω+) if and only if Ω + is vanishing Reifenberg flat, Ω + and Ω - have joint big pieces of chord-arc subdomains, and the inner unit normal of Ω + has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.
AB - Let Ω +⊂ Rn+1 be an NTA domain and let Ω -= Rn+1\ Ω +¯ be an NTA domain as well. Denote by ω+ and ω- their respective harmonic measures. Assume that Ω + is a δ-Reifenberg flat domain for some δ> 0 small enough. In this paper we show that logdω-dω+∈VMO(ω+) if and only if Ω + is vanishing Reifenberg flat, Ω + and Ω - have joint big pieces of chord-arc subdomains, and the inner unit normal of Ω + has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.
UR - http://www.scopus.com/inward/record.url?scp=85085054740&partnerID=8YFLogxK
U2 - 10.1007/s00526-020-01760-2
DO - 10.1007/s00526-020-01760-2
M3 - Article
AN - SCOPUS:85085054740
SN - 0944-2669
VL - 59
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 3
ER -